Point-based rigid registration is commonly used for image-guided systems. One set of points is to be registered to another set of corresponding points by means of a rigid transformation of the first set. Surgical guidance systems based on pre-operative images, such as CT or MRI, typically employ a tracking system to map points from image to the physical space of the operating room. For neurosurgery and ear surgery, because of the rigidity of the skull, the point mapping is typically a rigid transformation. The transformation is usually based on fiducial markers that are attached to the head before imaging and remain attached until the procedure begins. In general, a fiducial point set is obtained by localizing each fiducial marker both in the image and in the operating room. Then, a least-squares problem is solved to register the image points to their corresponding physical points, and the result is the rigid transformation. Typically, during such registration processes, fiducial localization error (FLE) results in registration errors, and a least-squares approach is commonly used to obtain the transformation that minimizes this error in fiducial alignment. The least-squares solution has a closed form when FLE is isotropic, but a closed form is generally unavailable in the case of anisotropic FLE, for which anisotropic weighting is required. With fiducial markers that are larger than a voxel, FLE in the image space can be made somewhat isotropic, but FLE in physical space may suffer from severe anisotropy due to tracking. For example, physical positions are often acquired via optical systems, whose localization error is highly anisotropic. Furthermore, the positions are often reckoned relative to a coordinate reference frame (CRF) that is rigidly attached to the patient. The use of a CRF enables patient movement relative to the tracking system during the procedure, but it tends to exacerbate the anisotropy.